4292
Macromolecules 1990, 23, 4292-4298
A Kinetic Description of Diffusion-Controlled Intramolecular Excimer and Exciplex Formation Guojun Liu and J. E. Guillet' Department of Chemistry, University of Toronto, Toronto, Ontario, Canada M5S 1 A1 Received July 20, 1989; Revised Manuscript Received February 23, 1990
ABSTRACT A new scheme is proposed for diffusion-controlledexcimer formation between polymer chain ends. Unlike the Birks scheme,' the new scheme makes no use of rate expressions for excimer formation and dissociation. Instead, excimer formation is described as the natural consequence of chain end diffusion. Computer simulationbased on the new scheme yields monomer and excimer decay curves similar to the ones experimentally observed. When used for intramolecular excimer formation kinetics, a convenient and sensitive method can be developed from this theory for measuring the diffusion coefficients, D, of polymer ends, the free energy, to, for excimer formation, root-mean-squareend-to-enddistances, R., of polymer chains, and fluorescence lifetimes of excimers, 'TD.
I. Introduction The kinetics of excimer and exciplex formation has previously been described by the well-known Birks scheme:'
M
D
However, the general use of the Birks scheme in describing intramolecular excimer formation has a few disadvantages. (1) kl(t) do not necessarily reach their asymptotic values fast enough in the time domain in which the experiment is performed. (2) It makes unnecessary use of rate equations. The reaction of excimer formation is diffusion-controlled, and it is sufficient to describe the process by diffusion equations. Due to the unnecessary use of rate laws, time-dependent rate constants arise sometimes that are in contradiction to the conventional definition of a constant. (3) The kinetic description of the system becomes unnecessarily complex when one introduces the time-dependent rate constant kl(t).22 This paper presents a simple and straightforward scheme for describing intramolecular excimer formation kinetics. In this approach, no rate equations will be used for describing excimer formation and dissociation. Since excimer formation is assumed to be diffusion-controlled, excimer formation is described as arising from the natural diffusion motion of the chain ends.
where k~ and kD are rate constants for self-deactivation processes of the excited monomer M* and excimer D*; kl and k-1 are rate constants for the interchange reactions between M* and D*. The above scheme has been widely used to describe both intermolecular2*4and intram0lecular5>~excimer or exciplex formation. When used to describe excimer formation between two reactive groups connected by a long polymer chain undergoing Brownian motion, the intramolecular rate constant kl can be calculated for different polymeric dynamic models using the .general theory for diffusion11. Theory controlled reactions first proposed by Wilemski and Excimer o r Exciplex. A molecule M in an electroniF i ~ m a n and ~ - ~subsequently modified by others.10-18 cally excited state M* may be a very polarizable species According to Wilemski and F i ~ m a n , rates ~ - ~ of reactions and interacts with other polar or polarizable species. If between end groups of a polymer chain can be depicted M* interacts with M and forms a collision complex D*that as the rate a t which one end of a polymer chain enters a is more stable than M* M, D*is called an excimer. If sink, such as a reaction sphere of radius Rg,selected around M* forms a collision complex with a different molecule the other end group. The rate constant kl has been found N, the complex formed is called an exciple^.^^ Diffusion to be time-dependent. At very short times, the reaction and kinetic equations are developed below for the system is controlled by the intrinsic rate kCo, where k is the based on a discussion of excimer formation. The equations intrinsic second-order rate constant and CO is the are equally applicable to exciplex formation kinetics, equilibrium sink concentration, the portion of polymer provided that there are no accompanying electronchains that possess the configurations enabling the end transfer processes. groups to react at time t = 0.l8 In the long-time limit ( t a),the reaction can be described by a uniquely defined The interaction between M* and M is orientation- and rate constant k l ( t - - ) that is independent of k , the distance-dependent. For planar molecules such as pyrene intrinsic second-order rate constant, if the reaction is (Py),parallel pairs of Py and Py* with overlapping center diffusion-controlled.'s Rather, 1/k l ( t - m ) is characteristic axes are energetically more favorable orientations.24 Since of the longest relaxation time of the polymer chain that the relative rotational motion between M* and M is, in connects the reactive groups. In the intermediate times, general, much faster than the relative diffusional motion k l ( t ) has a complicated time dependence.ls between them,25it can be assumed that most of these M* In fortunate cases, k l ( t ) in certain systems actually and M pairs that are brought together by the relative reaches its asymptotic value kl(t---) very rapidly (- 1ns) diffusion between them are in the energetically more and the experimental determination of kl(t-m) by favorable parallel orientations with center axes of M* and M overlapping. For pyrene molecules, Py* and Py, studying excimer formation reactions between groups attached to polymer chain ends has been p o ~ s i b l e . l ~ - ~ ~approaching one another in the parallel fashion, the attractive part of the interaction potential has been shown These studies greatly contributed to the understanding of the dynamic behavior of polymer chains. to be inversely proportional to the third power of the in-
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0024-9297/90/2223-4292$02.50/0
0 1990 American Chemical Society
Macromolecules, Vol. 23, No. 19, 1990
Diffusion-Controlled Excimer Formation 4293
terplanar separation distance,' or
u , a~i i r 3
correction factor:
(2)
In this paper, kinetic equations for excimer formation are derived assuming that the interaction potential between M* and M is described by the Lennard-Jones equation:26
P(r) = y-'G(r) exp[-U(r)/kTI where k T is the thermal energy. yl, a normalization factor, is given by
u(r)= -c0[2(r0/r)6- (
